There are differences between the Camping merit badge requirements and the Second Class / First Class campout requirements.
Camping merit badge
9. Show experience in camping by doing the following:
(a) Camp a total of at least 20 nights at designated Scouting activities or events. One long-term camping experience of up to six consecutive nights maybe applied toward this requirement. Sleep each night under the sky or in a tent you have pitched. If the camp provides a tent that has already been pitched, you need not pitch your own tent.
The BSA’s program / advancement team has provided some additional guidance (Ask the Expert: What is (and what isn’t) a camping night for the Camping MB?) on what qualifies as a night of camping for purposes of the Camping merit badge:
“The intent of the requirement is to camp overnight in a tent or under the stars,” LoVecchio says. “This means sleeping overnight in building/structure does not meet the intent of the requirement.”
Is sleeping in a cave the same as sleeping in a building / structure? Or is it more similar to sleeping “under the sky”? Does it make a difference that the structure (caves) were made by nature - not man-made? Is the intent of the requirement to ensure that the Scouts have some kind of exposure to the elements (possibility of getting rained on, changes in temperature, insects, etc.)? This is why I think that different Merit Badge Counselors might reasonably come to different conclusions. I recommend consulting with your district / council Advancement Chair for advice.
The requirements for Second Class / First Class 1a. are different:
…On campouts, spend the night in a tent that you pitch or other structure that you help erect, such as a lean-to, snow cave, or tepee.
The intent of Second Class / First Class requirement 1a. seems to be to require Scouts to gain some experience in pitching tents or erecting some kind of shelter. According to the original poster, the Scouts are not pitching tents or erecting other structures, so sleeping in caves would not appear to qualify as a “campout” for purposes of SC / FC requirement 1a. However, it would still count as a troop or patrol “activity”.